Fixed Point Property Research Articles

The topological structures of the spaces of copulas and subcopulas

The (sub)copulas play a central role in statistics for capturing the stochastic dependence structure of random variables. A subcopula is a map from the Cartesian product of two closed subsets of the unit closed interval containing zero and one to the unit closed interval, which is grounded, 2-increasing and has uniform margins. A copula is a subcopula whose domain is the Cartesian product of the unit closed interval and itself. The uniform metric on the set of all copulas can be extended to a metric on the set of all subcopulas. In this work, we investigate topological structures of these spaces by using tools from infinite-dimensional topology. Our main result is as follows: there exists a homeomorphism from the space of subcopulas onto the Hilbert cube which sends the space of all copulas onto an end of the Hilbert cube and sends the space of exchangeable copulas onto an end of the end of the Hilbert cube. As a consequence, we obtain that the spaces of subcopulas, copulas and exchangeable copulas are homeomorphic to the Hilbert cube and hence have the fixed point property.

A new approach to fixed point property of nonexpansive multivalued mappings

A new approach to fixed point property of nonexpansive multivalued mappings

Weak fixed point property for a Banach lattice of some compact operator spaces

Weak fixed point property for a Banach lattice of some compact operator spaces

Fixed points of a family of mappings and equivalent characterizations

In the present paper we prove a fixed point theorem for a one parameter family of contractive self-mappings, of a complete metric space or a complete b-metric space, each member of which has a unique fixed point that is also the unique common fixed point of the family; the mappings may be continuous or discontinuous at the fixed point. We also prove that under the assumption of a weaker form of continuity the fixed point property for mappings satisfying the contractive conditions employed by us implies completeness of the underlying space. The characterization of completeness obtained by us not only contains Subrahmanyam?s theorem on characterization of completeness as a particular case but also extends it to b-metric spaces. Results on contractive mappings with discontinuity at the fixed point have found applications in neural networks with discontinuous activation function (e.g. Ozgur and Tas [19, 20]).

The fixed point property of quasi-point-separable topological vector spaces

The fixed point property of quasi-point-separable topological vector spaces

A fixed point theorem in strictly convex $ b $-fuzzy metric spaces

<abstract><p>The main motivation for this paper is to investigate the fixed point property for non-expansive mappings defined on $ b $-fuzzy metric spaces. First, following the idea of S. Ješić's result from 2009, we introduce convex, strictly convex and normal structures for sets in $ b $-fuzzy metric spaces. By using topological methods and these notions, we prove the existence of fixed points for self-mappings defined on $ b $-fuzzy metric spaces satisfying a nonlinear type condition. This result generalizes and improves many previously known results, such as W. Takahashi's result on metric spaces from 1970. A representative example illustrating the main result is provided.</p></abstract>

The topological structures of spaces of fuzzy n-cell numbers and fuzzy n-ellipsoid numbers

As two special types of the general n-dimension fuzzy numbers, fuzzy n-cell numbers and fuzzy n-ellipsoid numbers play an important role in the representations of imprecise multichannel digital information. Denote the families of all n-dimension fuzzy numbers, fuzzy n-cell numbers and fuzzy n-ellipsoid numbers, by K(Rn), L(Rn) and E(Rn), respectively. In this paper, we study the topological structures of the sets L(Rn) and E(Rn) with the topology induced by the supremum metric D∞. Our main result is as follows: for every n>1, there exist homeomorphisms H1,H2:(K(Rn),D∞)→[0,+∞)×ℓ2(c) such that H1(L(Rn),D∞)=H2(E(Rn),D∞)={0}×ℓ2(c), where ℓ2(c) is a non-separable Hilbert space whose weight is the cardinality of the set of all real numbers. As a consequence, the spaces (L(Rn),D∞) and (E(Rn),D∞) are homeomorphic to the Hilbert space ℓ2(c), and hence do not have the fixed point property.

On Nonexpansive and Expansive Semigroup of Order-Preserving Total Mappings in Waist Metric Spaces

In this paper, we introduce nonexpansive and expansive semigroup of order-preserving total mappings (ONTn) and (OETn), respectively, to prove some fixed point theorems in waist metric spaces. We examine the existence of mappings that satisfy the conditions ONTn and OETn. We also prove that every semigroup of order-preserving total mappings OTn has fixed point properties and that the set of fixed points is closed and convex. The present study generalised many previous results on semigroup of order-preserving total mappings OTn. Efficacy of the results was justified with some practical examples.

Higher-order concepts for the potential infinite

The lambda calculus can be seen as a language that connects mathematics with computer science. However, the models used in both disciplines have different basic properties. In computer science these models are mainly based on domain theory with its partial functions and its fixed point property. In mathematics the models use total functions and relations and allow the integration of logic. Typically, mathematical models are based on set theory with its actual infinite sets, which are incongruous with computer science.We will present a dynamic model that is based consequently on the potential infinite. It uses total functions, whereby functions and the function spaces are seen as interdependent, increasing finite entities. It moreover allows the integration of higher-order logic (to be demonstrated in a separate paper), provided the universal quantifier is interpreted in a specific, dynamic way. Such a model can serve as a common denotational semantics of the lambda calculus for both, mathematics and computer science.

Fixed point structures on a set-mapping pair and cartesian product

Abstract In this paper we study the following problem (Problem 4.2 in, I.A. Rus, Sets with structure, mappings and fixed point property: fixed point structures, Fixed Point Theory 23, No. 2 (2022), 689-706): Let (𝒰, 𝒮, M) be a fixed point structure on (𝒰, M). We suppose that (𝒰, M) is with cartesian product. In which conditions, we have that: X,Y∈𝒮⇒X×Y∈𝒮? Some results in terms of exponential with fixed point property are also given. The basic results are illustrated by examples.

Linear isomorphic spaces of Cesàro–Nörlund operator, their duals and matrix transformations

Abstract In this article, we introduce and study sequence spaces of Cesàro–Nörlund operators of order n associated with a sequence of Orlicz functions. We obtain some topological properties and Schauder basis of these sequence spaces. Moreover, we compute the α-, β- and γ-duals and the matrix transformations of these newly formed sequence spaces. Finally, we prove that these sequence spaces are of Banach–Saks type p and have a weak fixed-point property.

Abelianization and fixed point properties of units in integral group rings

AbstractLet G be a finite group and the unit group of the integral group ring . We prove a unit theorem, namely, a characterization of when satisfies Kazhdan's property , both in terms of the finite group G and in terms of the simple components of the semisimple algebra . Furthermore, it is shown that for , this property is equivalent to the weaker property (i.e., every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property , denoted . More precisely, it is described when all subgroups of finite index in have both finite abelianization and are not a nontrivial amalgamated product. A crucial step for this is a reduction to arithmetic groups , where is an order in a finite‐dimensional semisimple ‐algebra D, and finite groups G, which have the so‐called cut property. For such groups G, we describe the simple epimorphic images of . The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups of . These groups are well understood except in the degenerate case of lower rank, that is, for with an order in a division algebra D with a finite number of units. In this setting, we determine Serre's property FA for and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its ‐rank.

Approximating a Common Solution of Monotone Inclusion Problems and Fixed Point of Quasi-Pseudocontractive Mappings in CAT(0) Spaces

In this paper, we aimed to introduce a new viscosity-type approximation method for finding the common fixed point of a class of quasi-pseudocontractive mapping and a system of monotone inclusion problems in CAT(0) spaces. We proved some fixed-point properties concerning the class of quasi-pseudocontractive mapping in CAT(0) spaces, which is more general than many other mappings such as nonexpansive, quasi-nonexpansive, pseudocontractive and demicontractive mappings which have been studied by other authors. A strong convergence result is proved under some mild conditions on the control sequences and some numerical examples were presented to illustrate the performance and efficiency of the proposed method.

Weakly almost periodic functions invariant means and fixed point properties in locally convex topological vector spaces

Weakly almost periodic functions invariant means and fixed point properties in locally convex topological vector spaces

"A case study in set-mapping pair theory: the set-mapping pair bijections"

"In a recent paper (I. A. Rus. Sets with structure, mappings and fixed point property: fixed point structures. Fixed Point Theory 23 (2022), No. 2) the author introduced, amongst others, the following notions: set-mapping pair and set-mapping pair bijection. In this paper we study the set-mapping pair bijections in connection with the isomorphisms structure, and their impact on the fixed point theory in a set-mapping pair. Some open problems are also formulated"

Sets with structure, mappings and fixed point property: fixed point structures

Sets with structure, mappings and fixed point property: fixed point structures

On contractive mappings in bv(s)-metric spaces

The major motives of this paper are to study different types of contractive mappings and also to answer an open question of Garai et al. [The contractive principle for mappings in $b_v(s)$-metric spaces, arXiv:1802.03136]. We first set up some fixed point results associated with two types of contractive mappings in $b_v(s)$-metric spaces and then we give an answer, in positive, to the open question. Most importantly, we characterize the completeness of a $b_v(s)$-metric space via fixed point property of a certain type of contractive mappings. Our results extend and generalized several important results in the literature.

Testing for the Rayleigh Distribution: A New Test with Comparisons to Tests for Exponentiality Based on Transformed Data

We propose a new goodness-of-fit test for the Rayleigh distribution which is based on a distributional fixed-point property of the Stein characterization. The limiting null distribution of the test is derived and the consistency against fixed alternatives is also shown. The results of a finite-sample comparison is presented, where we compare the power performance of the new test to a variety of other tests. In addition to existing tests for the Rayleigh distribution we also exploit the link between the exponential and Rayleigh distributions. This allows us to include some powerful tests developed specifically for the exponential distribution in the comparison. It is found that the new test outperforms competing tests for many of the alternative distributions. Interestingly, the highest estimated power, against all alternative distributions considered, is obtained by one of the tests specifically developed for the Rayleigh distribution and not by any of the exponentiality tests based on the transformed data. The use of the new test is illustrated on a real-world COVID-19 data set.

Approximate fixed point theorems for weak contractions on neutrosophic normed spaces

In this paper, we define concept of approximate fixed point property of a function and a set in Neutrosophic normed space. Furthermore, we give Neutrosophic version of some class of maps used in fixed point theory and investigate approximate fixed point property of these maps.

Set Invariant Means and Set Fixed Point Properties

In this paper, we introduce a concept of fixed point property for a semigroup $S$ called $A$-fixed point property, where $A$ is a non-empty subset of $S$. Also, the relationship between $A$-amenability and $A$-fixed point property is investigated.